# Statements in logic

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```					  Logic
In Part 2 Modules 1 through 5, our topic is symbolic logic.

We will be studying the basic elements and forms that
provide the structural foundations for critical reasoning.

Symbolic logic is a topic that unites the sciences and the
humanities.

Researchers in logic may come from philosophy,
mathematics, linguistics, or computer science, among
other fields.
Statements in logic
In logic, a statement or proposition is a declarative
sentence that has truth value.
When we say that a sentence has truth value, we mean that
it makes sense to ask whether the sentence is true or
false.

“Today is Monday” is a statement.
“1 + 1 = 3” is a statement.
Quantifiers and categorical statements
In logic, terms like “all,” “some,” or “none” are called
quantifiers.

A statement based on a quantifier is called a quantified
statement or categorical statement.

“All bad hair days are catastrophes.”
“No slugs are speedy.”
“Some owls are hooty.”
are examples of quantified or categorical statements.
Categories
Quantified or categorical statements state a relationship
between two or more classes of objects or categories.

In the previous examples,
catastrophes
slugs
speedy (things)
owls
hooty (things)
are all categories.
Existential statements
A statement of the form “Some A are B” or “Some A aren’t B”
asserts the existence of at least one element (in logic,
“some” means “at least one”).

Categorical statements having those forms are called
existential statements.

“Some owls are hooty”
“Some wolverines are not cuddly”
are examples of existential statements.
Existential statements
“Some owls are hooty” asserts that there exists at least one
thing that is both an owl and hooty.
That is, the intersection of the categories “owls” and “hooty
things” is not empty.

We can convey that information by making a mark on a Venn
diagram. We place an “X” in a region of a Venn
diagram to indicate that that region must contain at
least one element.
Diagramming existential statements
Diagramming existential statements
The existential statement “Some wolverines are not cuddly”
asserts that there must be at least one element who is a
wolverine (W) but is not cuddly (C ).
Universal statements
“All bad hair days are catastrophes”
“No slugs are speedy”
are examples of universal statements.
Negative universal statements
A statement of the form “No A are B” is called negative
universal.
It asserts that there is no element in both category A and
category B at the same time.

In other words, “No A are B” asserts that categories A and B
are disjoint, which means that the intersection of the two
categories is empty.

“No slugs are speedy” is a negative universal statement.
Diagramming negative universal statements
In logic, we use shading to indicate that a certain region
of a Venn diagram is empty (contains no elements).

The negative universal statement “No slugs are speedy”
asserts that the region of the diagram where “Slugs” and
“Speedy things” intersect must be empty.
Diagramming negative universal statements
Positive universal statements
A statement of the form “All A are B” is called positive
universal.
It asserts that there is no element in category A that isn’t also
in category B.

“All bad hair days are catastrophes” is an example of a
positive universal statement.
Diagramming positive universal statements
The positive universal statement “All bad hair days are
catastrophes” asserts that it is impossible to be a bad
hair day (B) without also being a catastrophe (C).

This means that the region of the diagram that is inside B
but outside C must be empty.
Diagramming positive universal statements
Interpreting Venn diagrams in logic
We will use Venn diagrams (typically three-circle diagrams) to
convey the information in propositions about relationships
between various categories.
In logic, when a region of a Venn diagram is shaded, this
tells us that that region contains no elements.
That is, a shaded region is empty.
Suppose that we are presented with the marked Venn
diagram shown below and on the following slides. We
should be able to interpret the meaning of the marks on
the diagram.
An “X” means “something is here…”
In logic, when a region of a Venn diagram contains an
“X”, this tells us that that region contains at least one
element.
An “X” means “something is here…”
In logic, when an “X”, appears on the border between
two regions, this tells us that there is at least one
element in the union of the two regions, but we
are not certain whether the element(s) are in the
first region, the second region, or both regions.
No marking means “uncertain…”
In logic, when a region of the Venn diagram contains no
markings, it is uncertain as to whether or not that
region contains any elements.
Example
Suppose we will use a three-circle Venn diagram to convey information
about the relationships between these three categories: Angry apes
(A); Blissful baboons (B); Churlish chimps (C).
Select the diagram whose markings correspond to “No blissful baboons
are angry apes.”
Assume that we do not know of any other relationships between categories.
Solution
Select the diagram whose markings correspond to “No B are A.”
According to the proposition “No B are A,” it must be impossible for an
element that is in category A to also be in category B. This means that
the intersection of circles B and A must be empty (that is, shaded).
This is what is shown in choice B below. The correct choice is B.
More exercises
For tutorials on diagramming categorical propositions, see The DIAGRAMMER on